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Mathematical Models for Blood Coagulation

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Fluid-Structure Interaction and Biomedical Applications

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

This chapter presents an overview and introduction to blood coagulation models. The historical exposure of the development of classical coagulation modeling theories is followed by a basic overview of blood coagulation biochemistry. The recent developments of cell-based models are explained in detail to demonstrate the current shift from the classical cascade/waterfall models. This phenomenological overview is followed by a survey of available mathematical concepts used to describe the blood coagulation process at various spatial scales including some of the related biophysical phenomena. A comprehensive survey of basic literature is provided for each of these topics.

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Notes

  1. 1.

    Pregnancy may increase the risk of thrombosis in various ways. The swollen uterus can compress pelvic vessels, reducing blood circulation in the legs. Hormonal changes can also produce hypercoagulability by increasing the concentration in blood of pro-coagulant factors and reducing the concentration of anticoagulant factors. A more serious condition sometimes related to pregnancy is the Antiphospholipid Antibody (or sticky blood) Syndrome, due to the autoimmune production of antibodies against a cell membrane substance called phospholipid causing platelets aggregation.

  2. 2.

    The most famous physician of ancient Egypt was Imhotep (a semi-divine character, he lived during the twenty-seventh century bc and is supposed to be the legendary author of this papyrus). The practice of mummifying corpses must have taught much to Egyptians about the human body, but the papyrus (mainly dealing with wounds healing) can hardly be considered a scientific document and the suggested remedies could easily be fatal to the patients because they could produce infections.

  3. 3.

    He was the author of the humoral theory, according to which four humors (blood, phlegm, black bile, and yellow bile) had to be in a proper balance in healthy individuals. The theory, somehow anticipated by Alcmaeon of Croton (fifth century bc), parallels the contemporary claim by Empedocles that four elements (air, water, fire, earth) are the basic constituents of the world and which may have had a much older origin. Hippocrates tremendous authority (and the immense reputation of Galen, who took his legacy to the Roman world and passed it on to the next era) prevented the development of medicine on a scientific basis for centuries, thanks to the blindness of his followers. The humoral theory found its way through Islamic medicine: the Persian Avicenna (Ib Sı̄nā, 980–1037) based his Canon of Medicine (1025) on Hippocrates’ and Galen’s theories. It was instead opposed by another Persian, Razi (Muhammad ibn Zakariyā Rāzı̄, 865–925) an eclectic scientist, very famous in his times, who explicitly questioned several of Galen claims on the basis of his own experimental observations.

  4. 4.

    He adopted and propagated Hippocrate humoral theory, adding his own theory of four temperaments (choleric, melancholic, sanguine, phlegmatic), resulting from the combinations of the humors with four qualities of (cold, warm, moisty, dry). He sketched an erroneous scheme of the circulatory system. We had to wait until the famous treatise Exercitatio anatomica de motu cordis et sanguinis in animalibus (1628) by William Harvey (1578–1657) for a correct systematic description of blood circulation (limited to great vessels: microcirculation was a later discovery). It is worth mentioning at this point the important contributions later given by the English eclectic scientist Stephen Hales (1677–1761), who determined the blood volume, the heart output and who first measured arterial blood pressure. For completeness we recall the revolutionary work of Andreas Vesalius (Latinized from Andries van Wesel) (1514–1564), who opened a new era in physiology. It is interesting to note that Vesalius studied Rāzı̄’s books and that he based his famous treatise De humani corporis fabrica libri septem (1543) on direct observation of dissected human bodies. He pointed out several of Galen’s mistakes (particularly in the description of circulatory system), indifferent to the harsh criticism of Galen’s followers.

  5. 5.

    Explanations given to pregnancy or post-partum related limb swelling by various authors during the seventeenth and eighteenth centuries, largely based on humors look today simply ridiculous. See the paper [13].

  6. 6.

    Such a fibrous component was isolated much later by Marcello Malpighi (1628–1694), the Italian physician famous above all for his studies on kidneys.

  7. 7.

    A new course in the medical studies was set by the book The Philosophical Principles of Medicine (1725) by Thomas Morgan.

  8. 8.

    Cells were first observed at the microscope by the physicist Robert Hooke (1635–1703) (the founder of the theory of elasticity) in a thin sample of cork (1665). He did not know what cells were, but he called them that way because of their particular and regular arrangement in the sample, resembling the one of monks cells.

  9. 9.

    Malpighi first described RBC as fat corpuscles (1663). Actually RBCs had been observed earlier (1658) by the Dutch Jan Swammerdam (1637–1680). Malpighi was also the discoverer of capillaries (1661).

  10. 10.

    It is really amazing that a well-known scientist like the Swedish Robin Fåhraeus (1888–1968), still quoted today, for instance, in the field of blood rheology, believed to have found a confirmation of Galens theories on the basis of the observation that blood coagulates in four layers with different colours, corresponding to the famous humors [121].

  11. 11.

    Virchow described the mechanism of thromboembolism [240], a phenomenon that was by no means clear at his time (inflammation was considered by many physicians the real cause of thrombosis: this was the subject of a famous dispute with the French pathologist Jean Cruveilhier). Curiously, he did not formulate the famous triad which for some reason found a firm place in the literature much after his death (apparently not before 1950!). See the interesting review [17].

  12. 12.

    Circumcision is a very old practice, already found in the ancient Egypt and that was widely adopted also in the Islamic world. Its origin in ancient Egypt was probably as an initiation practice to religious offices. The Book of the Dead describes self-circumcision by the sun-god Ra: Blood fell from the phallas of Ra after he had finished cutting himself.

  13. 13.

    In numbers, 1 over 10,000 men is hemophilic. The probability that a woman is hemophilic is the square of that number.

  14. 14.

    In this connection the name of the eminent French hematologist Georges Hayem (1841–1933) has to be remembered as one of the founders of modern hematology. He performed the first count of platelets. In 1882 he illustrated the effects of thrombocytopenia (low platelets count).

  15. 15.

    Large molecules like proteins have specific sites which are engaged in specific reactions.

  16. 16.

    All data concerning human blood are subjected to large variations, according to sex, body weight, and health conditions.

  17. 17.

    AMP, ADP, ATP contain 1 (Mono-), 2 (Di-), 3 (Tri-) atoms of phosphorus and they are obtained in that sequence by addition of a P atom (a process called phosphorylation). ATP has a vital importance in cells metabolism.

  18. 18.

    This condition can be produced by arterial stenosis, possibly as a consequence of clotting itself, or due to the mechanical action of implanted devices (rigid artificial heart valves).

  19. 19.

    Platelet Factors 1–3 actually regulate interactions with the Coagulation Factors IIa (thrombin), V, X.

  20. 20.

    Ecto-enzymes act at the exterior of cells. For more details about this enzyme see, [87].

  21. 21.

    Though there is some disagreement on the exact meaning of this name, it is very frequently attributed to TF.

  22. 22.

    Serine proteases are a large class of enzymes including the amino acid serine. The list of serine proteases is impressively long. See http://biochem.wustl.edu/~protease/ser_pro_help.html.

  23. 23.

    Simultaneous and independent discoveries had produced a great confusion in nomenclature.

  24. 24.

    Actually there were many more names: see [208] for a complete list.

  25. 25.

    Fibrinogen is not just the precursor of Fibrin, but it has also other specific functions, illustrated in this chapter. It is also known to stimulate RBCs aggregation (forming the so-called rouleaux), a phenomenon of some importance in blood rheology.

  26. 26.

    In order to prevent coagulation and keep blood flowing through the wound it produces, the Hirudo Medicinalis (leech) secretes Hirudin, a natural and very effective inhibitor of thrombin.

  27. 27.

    Stephen Christmas was the first patient diagnosed with FIX deficiency (hemophilia B) (1952) at the age of five. He died in 1993 by AIDS. Many of the transfusion-dependent patients were infected by the HIV virus before blood screening became obligatory. A case which became emblematic was the one of Ryan Wayne White, affected by hemophilia A, who became discriminated when he was diagnosed with AIDS. He died still a teenager in 1990.

  28. 28.

    Named after the patients Rufus Stuart and Audrey Prower.

  29. 29.

    Named after Ratnoff’s patient John Hageman (1955).

  30. 30.

    Laki and Lorand suggested its existence in 1948 [138].

  31. 31.

    A very important function of Protein S in the organism is to facilitate phagocytosis of apoptotic cells by macrophages. Discovered in 1979 in Seattle, takes its name after that city.

  32. 32.

    Denominated after the German name Koagulationvitamin. Discovered in the 1930 a Nobel prize was attributed in 1943 for studies on it, though its real action in the coagulation process became clear only in the 1970.

  33. 33.

    Patented in 1948 as a rat poison and used as anticoagulant for humans since 1954. It was isolated in 1941 by a group at the University of Wisconsin after a 6-year work investigating a widespread hemorrhagic disease that affected cattle in the USA, the so-called sweet clover disease (the research was funded by WARF, i.e., Wisconsin Alumni Research Foundation). See [248].

  34. 34.

    Already in the 1950 it was known that deficiency of vWF was accompanied by a deficiency of FVIII (see [208]).

  35. 35.

    The symbol X-ase is sometimes used.

  36. 36.

    Also referred to as Williams Factor or Flaujeac Factor.

  37. 37.

    Kininogens are proteins which are precursors of kinins (see next footnote), such as bradikinin and kallidin, which are vasodilator.

  38. 38.

    Here we refer to Plasma Kallikrein, distinct from the numerous group of Tissue Kallikreins, which are enzymes performing various actions. Discovered in 1934, it was named after the Greek words kalli (sweet, in this context) and krein (flesh) referring to pancreas tissue. Plasma Kallikrein (like some of its tissue analogs) liberates kinins from the kininogens. The so-called kinin-kallikrein system has a role in regulating blood pressure, owing to the vasodilation action.

  39. 39.

    First isolated in the urine.

  40. 40.

    Also Kallikrein and FXIIa can activate plasminogen.

  41. 41.

    AT I–IV are also found in the literature, with specific targets.

  42. 42.

    Discovered in 1918 [109], though isolated in 1916 in canine liver tissue [167]. There has been some controversy about heparin discovery (see [248] and [162]]). It is a large polymer, also naturally produced by endothelial cells (as heparan sulfate). A side effect can be a strong reduction of platelets count (Heparin Induced Thrombocytopenia, HIT), see [128]. HIT can be sometimes observed in patients undergoing hemodialysis, during which heparin is supplied to prevent clotting (after passing through the dialyzer and before being returned to the patient, protamine sulfate is added, which neutralizes heparin’s action). Platelet Factor 4 contrasts the action of heparin on platelets.

  43. 43.

    Its deficiency leads to degradation of tissues, particularly in the lungs, causing emphysema. Smoke is believed to inactivate this serpin, thus causing additional damage to lungs.

  44. 44.

    PAI2 is detectable only in pregnant women, a fact that may justify the increased risk of thrombosis during pregnancy.

  45. 45.

    Some FVIIa can reach TF in nonvascular tissues even in the absence of a lesion [279], thus making FIXa and FXa accidentally available. However, coagulation does not start because it requires, for instance, the intervention of platelets, which are not available out of the bloodstream.

  46. 46.

    The proteins responsible for this regulatory action have a fundamental role in eventually halting the clot growth. They are inevitably produced at this initial stage too, but it is known that FVa inhibition by APC is far less efficient than on the surface of endothelial cells [106, 219]. One can wonder whether, besides the clotting confining action, the simultaneous slowing down of the initial process may have a precise aim, for instance letting the platelet plug become thicker.

  47. 47.

    The Latin word for purple. Purpura denotes spots in the range 3–10 mm, smaller spots are called petechiae, and those more extended are called ecchymoses.

  48. 48.

    The Latin equivalent of the Greek derived word idiopathic is sui generis. In this context it means of no specific origin.

  49. 49.

    Congenital deficiency accounts for a small fraction of TTP cases and is known as Upshaw–Schülman syndrome.

  50. 50.

    More common among Ashkenazi Jews.

  51. 51.

    Because immobilization is a frequent cause, DVT is also called the economy class syndrome, since many cases have been reported in passengers after long flights.

  52. 52.

    In that case it is known as Paget–Schrötter disease.

  53. 53.

    Not with fibrinolytic proteins (like tPA or UPA), because they could fragment rather than gradually dissolve the clot. Fibrinolytic therapies are instead used to attack arterial thrombosis in the heart or the brain.

  54. 54.

    Major veins are provided with valves preventing flux inversion, thus helping circulation in the presence of reduced pressure gradients.

  55. 55.

    Reduced oxygen concentration is more marked in valves, since, differently from veins and other blood vessel, they do not possess their own vessels (vasa vasorum).

  56. 56.

    This paper is an extensive study on the role of TF and of thrombin in promoting angiogenesis and contains a large bibliography. Excessive TF production may be accompanied by upregulated expression of VEGF (the angiogenic factor) and downregulated expression of thrombospondin 1 (see Sect. 7.3).

  57. 57.

    Clots are mostly originated in the left atrium and more precisely in an area called left atrial appendage.

  58. 58.

    See, e.g., http://courses.washington.edu/overney/NME498_Material/NME498_Lectures/\Reading_on_Adsorption_Kinetics.pdf.

  59. 59.

    The scheme is inspired to an original document held by Ratnoffs heirs (see [208]).

  60. 60.

    e.g., Nuclear Magnetic Resonance (NMR) spectroscopy, X-ray crystallography.

  61. 61.

    See [237] for historical evolution and future trends in MD simulations complexity with respect to computational power growth.

  62. 62.

    This can easily be applied to non-Newtonian fluids, if necessary. However, for blood flows we usually assume the fluid phase being the blood plasma which is a Newtonian fluid under physiological conditions.

  63. 63.

    approximated numerically in the simulations.

  64. 64.

    See, e.g., [192] for platelet simulations.

  65. 65.

    In a similar way as in Smoothed Particle Hydrodynamics (SPH) [146, 175] where the interpolation kernel is usually truncated to have a compact support. The SPH method differs significantly from many other particle methods because the equations of motion for the fictitious particles in SPH are derived directly from the partial differential equations of fluid mechanics by integration using an interpolation kernel [66].

  66. 66.

    We use the same notation of particles position vectors as in the description of the DPD method.

  67. 67.

    The net force acting on the particle i is \(\boldsymbol{f}_{i} =\sum _{i\neq j}\boldsymbol{f}_{ij}\).

  68. 68.

    It uses similar principles as the Monte Carlo method in sub-microscale models.

  69. 69.

    See [93] for the relation between this stochastic approach and continuous deterministic reaction rate equations.

  70. 70.

    This equation is also known as Smoluchowski coagulation equation [221].

  71. 71.

    The size can be, e.g., particle volume, mass, or dimension.

  72. 72.

    Also called breakage function.

  73. 73.

    A basis for this approach comes from atmospheric science where it was used to describe cloud formation [28].

  74. 74.

    See p. 555 for details.

  75. 75.

    The subscript Ia refers to the chemical notation for fibrin.

  76. 76.

    See p. 554 for details.

  77. 77.

    The dimensions are normalized using the vessel radius (half-diameter) R = 3. 1 mm.

  78. 78.

    We should clarify that the simplification introduced assigning a pivotal role to prothrombinase and summarizing in one equation the complex process leading to its production makes sense only in the framework of the normal physiological process. If we have to consider any type of pathology referring, e.g., to a defective or missing factor, the model has to incorporate the equations involving the dynamics related to that specific factor. In other words, the model is conceived in an elastic way, adapting the number of equations to the complexity that needs to be taken into account.

  79. 79.

    In contrast to a linear, continuous change in the previous model.

  80. 80.

    I.e. a homogeneous Neumann condition in the context of this model.

  81. 81.

    We don’t speak here about time-scales explicitly, however it is clear that each of the sub-models, depending on its resolved spatial scale, has an associated time scale that is able to treat (within a reasonable computational time and accuracy).

  82. 82.

    In contrast with spatially two- or three-dimensional models.

  83. 83.

    Square brackets are used to distinguish the concentrations of the corresponding chemicals from their names.

  84. 84.

    released from an injured vessel wall (Tissue Factor TF)

  85. 85.

    From the biochemical point of view the first term on the right-hand side of (7.73) has a dubious interpretation. Actually VIIa is present in small quantities but its production has to come from VII and is mediated by VIIa itself (positive loop), in addition to the Xa coming out at the initiation stage. All this is replaced by a constant stimulus α. Thus it should be kept in mind that this model is a shortcut. Actually, it bypasses the action of the tenase complex, which in turn involves VIIIa and IXa.

  86. 86.

    This paper by H.C. Hemker et al. is entitled Is there value in kinetic modeling of thrombin generation? No (unless…). This review presents a summary of models showing the number of reactions they take into account, using the number of rate constants, based on the amount of cited papers. The rather critical point of view from this paper is balanced by another paper in the same issue of the journal written by K.G. Mann, under an almost identical title Is there value in kinetic modeling of thrombin generation? Yes [159].

  87. 87.

    in one or more spatial dimensions

  88. 88.

    For homogeneous models the Initial Value Problem is solved subject to initial data. By adding spatial variability to concentrations, the Initial Boundary Value Problem has to be solved using both, initial and boundary conditions.

  89. 89.

    See [80] for details of derivation and use.

  90. 90.

    See the paragraph on Euler–Lagrangian Particle Tracking methods (ELPT) in the section on microscale coagulation methods.

  91. 91.

    Subject to appropriate initial and boundary conditions.

  92. 92.

    Being positive, e.g., in the fluid (blood) and negative in the solid (thrombus).

  93. 93.

    In Lagrangian particle tracking form.

  94. 94.

    Usually some kind of norm of the stress tensor. See, e.g., [214, 222].

  95. 95.

    Platelet does not get activated immediately, but only after a certain time t act.

  96. 96.

    The thrombus model employs a composite structure with an impermeable core (activated platelets and fibrin) and a permeable shell (fibrin cap).

  97. 97.

    The shear-rate is defined as \(\dot{\gamma }= 2\sqrt{\boldsymbol{D}: \boldsymbol{D}}\) where \(\boldsymbol{D} = (\nabla \boldsymbol{u} + \nabla \boldsymbol{u}^{T})/2\) is the symmetric part of velocity gradient.

  98. 98.

    The upper-convected derivative \(\stackrel{\triangledown }{\boldsymbol{M}}\) of a tensor \(\boldsymbol{M}\) is defined using the classical material time-derivative \(\dot{\boldsymbol{M}}\) and the symmetric resp. skew-symmetric parts of the velocity gradient \(\boldsymbol{D}\) resp. \(\boldsymbol{W}\) as \(\stackrel{\triangledown }{\boldsymbol{M}} =\dot{\boldsymbol{ M}} -\boldsymbol{WM} + \boldsymbol{MW} - (\boldsymbol{DM} + \boldsymbol{MD})\).

  99. 99.

    See [6] for the complete algorithm of platelet activation depending on the values of \(\mathcal{A}\).

  100. 100.

    The subscript κ p (t) is used to emphasize that the stretch is expressed with respect to natural (time-dependent) configuration κ p (t). This notation follows exactly the original papers [7, 12, 199] and [34] where the model has been introduced and used.

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Acknowledgements

The financial support for the present project was partly provided by the Czech Science Foundation under the Grant No.201/09/0917 and by the Portuguese Science Foundation under The Research Center CEMAT-IST and under the Project EXCL/MAT-NAN/0114/2012.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    Antonio Fasano

  3. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    Adélia Sequeira

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  1. Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague 1, Czech Republic

    Šárka Nečasová

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Bodnár, T., Fasano, A., Sequeira, A. (2014). Mathematical Models for Blood Coagulation. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0822-4_7

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